| L(s) = 1 | + 2-s − 2·3-s + 4-s + 3·5-s − 2·6-s − 2·7-s + 8-s + 9-s + 3·10-s − 2·12-s − 2·14-s − 6·15-s + 16-s + 3·17-s + 18-s − 2·19-s + 3·20-s + 4·21-s + 6·23-s − 2·24-s + 4·25-s + 4·27-s − 2·28-s − 3·29-s − 6·30-s − 2·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 1.34·5-s − 0.816·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.577·12-s − 0.534·14-s − 1.54·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.458·19-s + 0.670·20-s + 0.872·21-s + 1.25·23-s − 0.408·24-s + 4/5·25-s + 0.769·27-s − 0.377·28-s − 0.557·29-s − 1.09·30-s − 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.83252489815880, −14.49042309344208, −13.94677994858801, −13.26377326434779, −12.88001721218453, −12.68116405768178, −11.91005352521213, −11.39996492468646, −10.88722189589920, −10.40037802209197, −9.899430427464274, −9.352350021543841, −8.861586630504885, −7.913741763488150, −7.210051495446534, −6.590119907364614, −6.218999319275119, −5.721469874241977, −5.370241463284519, −4.743853447481495, −4.059205521172059, −3.056304258152787, −2.759838716650724, −1.738255377969612, −1.094801582720133, 0,
1.094801582720133, 1.738255377969612, 2.759838716650724, 3.056304258152787, 4.059205521172059, 4.743853447481495, 5.370241463284519, 5.721469874241977, 6.218999319275119, 6.590119907364614, 7.210051495446534, 7.913741763488150, 8.861586630504885, 9.352350021543841, 9.899430427464274, 10.40037802209197, 10.88722189589920, 11.39996492468646, 11.91005352521213, 12.68116405768178, 12.88001721218453, 13.26377326434779, 13.94677994858801, 14.49042309344208, 14.83252489815880
